Edge-magic Indices of (n, n – 1)-graphs

A graph G = (V, E) with p vertices and q edges is called edge-magic if there is a bijection f : E → {1, 2, …, q} such that the induced mapping f+ : V → Zp is a constant mapping, where f+ (u) ≡ ∑uv ∈ E f(uv) (mod p). A necessary condition of edge-magicness is p ∣ q(q+1). The edge magic index of a graph G is the least positive integer k such that the k-fold of G is edge-magic. In this paper, we prove that for any multigraph G with n vertices, n − 1 edges having no loops and no isolated vertices, the k-fold of G is edge-magic if n and k satisfy a necessary condition for edge-magicness and n is odd. For n even we also have some results on full m-ary trees and spider graphs. Some counterexamples of the edge-magic indices of trees conjecture are given.